Are CSPRNGs non deterministic?

/ / Published in Cybersecurity, EITC/IS/CCF Classical Cryptography Fundamentals, Stream ciphers, Stream ciphers, random numbers and the one-time pad

Cryptographically Secure Pseudorandom Number Generators (CSPRNGs) are a critical component in the field of cybersecurity, particularly within the realms of classical cryptography and stream ciphers. To address the question of whether CSPRNGs are non-deterministic, it is essential to consider the definitions, functions, and characteristics of CSPRNGs, as well as their distinction from true random number generators (TRNGs).

A CSPRNG is designed to generate sequences of numbers that approximate the properties of random numbers. However, unlike true random number generators, which derive randomness from physical processes, CSPRNGs are algorithmic and deterministic by nature. This deterministic property means that if the initial seed value and the algorithm are known, the entire sequence of numbers generated by the CSPRNG can be predicted and reproduced.

The primary purpose of a CSPRNG is to produce output that is indistinguishable from true randomness to any efficient statistical test. This property is important for cryptographic applications where predictability could lead to vulnerabilities. Despite their deterministic nature, CSPRNGs are constructed to be computationally infeasible to predict without knowledge of the seed.

To illustrate this concept, consider an example of a CSPRNG based on a cryptographic hash function. Suppose we use a well-known hash function like SHA-256 to generate pseudorandom numbers. The process might involve taking an initial seed value, hashing it to produce the first pseudorandom number, and then using this output as the seed for the next iteration. The sequence generated by this method is deterministic, as the same initial seed and hash function will always produce the same sequence. However, without knowledge of the initial seed, it becomes computationally infeasible to predict future values in the sequence, assuming the hash function is cryptographically secure.

The deterministic nature of CSPRNGs does not detract from their utility or security, provided that the seed is kept secret. In practice, securely generating and managing the seed is paramount. For instance, seeds can be derived from high-entropy sources such as TRNGs, which leverage physical phenomena like electronic noise, radioactive decay, or other quantum processes to produce truly random values. Once a secure seed is obtained, a CSPRNG can generate a vast sequence of pseudorandom numbers that can be used for various cryptographic purposes, such as key generation, initialization vectors, and nonces.

A noteworthy property of CSPRNGs is their resistance to "state compromise extensions." This means that even if an adversary learns part of the internal state of the CSPRNG, they should not be able to reconstruct past outputs or predict future outputs. This property is essential for maintaining the security of cryptographic protocols that rely on pseudorandom numbers.

To further elucidate the distinction between CSPRNGs and TRNGs, consider the one-time pad (OTP) encryption scheme. The OTP requires a key that is as long as the message to be encrypted and is used only once. For the OTP to be perfectly secure, the key must be truly random, which necessitates the use of a TRNG. In contrast, if a CSPRNG were used to generate the key, the deterministic nature of the CSPRNG could potentially compromise the security of the OTP if the seed were discovered.

In the context of stream ciphers, CSPRNGs are often employed to generate keystreams. A stream cipher encrypts plaintext by combining it with a pseudorandom keystream generated by the CSPRNG. The security of the stream cipher hinges on the unpredictability of the keystream, which, in turn, depends on the security of the CSPRNG and the secrecy of its seed. Examples of stream ciphers that utilize CSPRNGs include RC4, Salsa20, and ChaCha20.

The RC4 stream cipher, for instance, initializes its internal state with a key-scheduling algorithm (KSA) that takes a secret key as input. The KSA permutes an internal state array based on the key, and the permutation is then used to generate the pseudorandom keystream during the encryption process. The keystream generated by RC4 is deterministic, as it is entirely dependent on the initial key. However, the security of RC4 relies on the difficulty of determining the key from the keystream.

Salsa20 and ChaCha20 are modern stream ciphers that also utilize CSPRNG principles. These ciphers initialize their internal state with a key and a nonce, and then apply a series of cryptographic transformations to produce the keystream. The resulting keystream is deterministic, but without knowledge of the key and nonce, it is computationally infeasible to predict or reproduce.

While CSPRNGs are inherently deterministic, their design ensures that their output is indistinguishable from true randomness to any efficient statistical test, provided the seed remains secret. This deterministic property does not undermine their utility in cryptographic applications, as the computational infeasibility of predicting the output without the seed ensures their security. The distinction between CSPRNGs and TRNGs is important, with the latter being necessary for applications requiring true randomness, such as the one-time pad, and the former being suitable for generating pseudorandom sequences for stream ciphers and other cryptographic protocols.

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